Động lực phi tuyến và đồng bộ hóa hệ thần kinh - Luận án tiến sĩ Mark Kramer

Nonlinear dynamics and neural systems: synchronization and modeling by Mark Alan Kramer BA (Oberlin College) 2001 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Applied Science and Technology in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY Committee in charge: Professor Andrew J. Szeri, Chair Professor Edgar Knobloch Professor Robert T. Knight Fall 2005 UMI Number: 3210316 Copyright 2005 by Kramer, Mark Alan All rights reserved. UMI Microform 3210316 Copyright 2006 by ProQuest Information and Learning Company.

All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P. Box 1346 Ann Arbor, MI 48106-1346 The dissertation of Mark Alan Kramer is approved: Chair Date Date Date University of California, Berkeley Fall 2005 Nonlinear dynamics and neural systems: synchronization and modeling Copyright 2005 by Mark Alan Kramer 1 Abstract Nonlinear dynamics and neural systems: synchronization and modeling by Mark Alan Kramer Doctor of Philosophy in Applied Science and Technology University of California, Berkeley Professor Andrew J.

Szeri, Chair We study the electrical activity of the human cortex in two ways. First, we state seven cou- pling measures to analyze electroencephalogram and electrocortiogram time series. We apply these measures to simulated and observed data, and we use the measures to deduce changes in coupling induced by auditory stimuli and produced by dementia. Second, we define a mathematical model of the spatially averaged, mean-field cortical electrical activity recorded by the electroencephalograph and electrocortiograph.

We compare the model results with ictal electrocortical data collected from four human subjects, and we show that the observed and simulated results agree in two important ways. We use the model to develop three methods for controlling seizures through electrical stim- ulation and to suggest the physiological mechanisms — and points of leverage for therapies — of epilepsy. Szeri Dissertation Committee Chair i Contents List of Figures iii List of Tables xviii 1 Introduction 1 2 Coupling measures 7 2.2 Example: Henon map .3 Example: coupled Rössler oscillators .4 Example: Oscillatory Bursts .1 Bursting data versus noise .5 Application: auditory ECoG ERP data.6 Application: discrimination between healthy and demented subjects .1 Clinical Diagnosis and Data Collection .2 Methods of Analysis .2 Observational Data: ECoG Seizure Recordings .3 Model: Dimensionless SPDEs .4 Simulations: Dimensionless ODEs .1 Example: Dimensionless ODES at Pee 11 0
.2 Example: Dimensionless ODEs at Pee 548 066 .5 Simulation: Dimensionless SPDEs .7 Bifurcation control of the seizing cortex .3 Control of stochastic partial differential equations .8 Additional routes to seizure. 169 4 Conclusion 180 Bibliography 184 iii List of Figures 2.1 (a) The electric potential of one ensemble member recorded by one electrode in the ECoG ERP experiment we discuss in detail in Section 2.

The stimulus occurs at t 0 ms. Note the oscillatory burst between 30 ms and 160 ms. The asterisk marks a point on the oscillatory burst. We chose d 3 and τ 10 for illustrative purposes and project the 3-dimensional embedding onto the plane of the page.

The asterisk in this figure corresponds to the asterisk in (a). The five nearest neighbors to this point are marked by triangles. The nearest neighbors are temporally proximal to the fiducial point because the data set is short.2 (a) Ten ensemble members embedded using d 3 and τ 10. The thickest curve is the ensemble member shown in Figures 2.

The other nine en- semble members are difficult to distinguish. The point x k n
 is marked with an asterisk. (b) The local neighborhood of the point x k n
. The thickest curve is the the trajectory of xk n.

The point xk n
 is marked with an asterisk. The thin curves are trajectories of nine other ensemble members. The nearest ensemble neighbors are marked with triangles.3 (a) The first ensemble members of s k n (solid line) and r k n (dashed line) gener- ated from the unidirectionally coupled non-identical Henon map for 50 s n 200 s. The coupling for 100 s n 150 s is not obvious.

(b) The windowed cross- correlation (WCC) between sk n and rk n. We plot the center time of each window along the horizontal axis, the lag time along the vertical axis, and the value of the cross-correlation in linear greyscale with values greater than 0 8 in black, less than

0 8 in white, and near 0 0 in grey. All WCC figures follow this color scheme

unless otherwise indicated. The WCC reveals a strong correlation between s k n and rk n at zero lag for 100 s n 150 s.

(c) The windowed coherence (WC) be- tween sk n and rk n. We plot the center time of each window along the horizontal axis, the frequency along the vertical axis, and the value of the coherence in linear greyscale with values greater than 0 8 in black and near 0 0 in white. The coher-

ence between sk n and rk n is strong for all frequencies when 100 s n 150 s.4 Computation of the embedding parameters for the time series generated from the unidirectionally coupled non-identical Henon map. (a) The average mutual in- formation (AMI) of the concatenated s k n as a function of time lag.

No relative minimum exists for any lag. (b) The percentage of false nearest neighbors of the concatenated sk n as a function of the embedding dimension. The value asymp- totes to a small, positive number for dimension 4 and greater. This is due to the random initial values of sk n for each ensemble member.5 Synchronization measures applied to the unidirectionally  coupled non-identical  Henon map.

All of the measures are smoothed over a window of size 11 at each time point. All three measures increase during the interval of nonlinear coupling (100 s n 150 s) between the chaotic time series. (b) The time shifted synchronization measure T x n  η y  smoothed over a two-dimensional window of size 11 at each time point. Note that the horizontal and vertical axes show time along ensembles r k n and sk n , respectively.

In the contour plot, there are five evenly spaced contour levels,  ranging from 0. Unless defined otherwise, all T x n  η y  figures follow this grey-scale scheme. The diag- onal line in the figure corresponds to the location of zero time lag. The contour plot shows synchronization occurs with time shift η 0 during the time interval 100 s n 150 s.

(c) The windowed phase synchronization. We plot the center time of each window along the horizontal axis, the phase (in radians) along the vertical axis, and the value of the phase synchronization in linear greyscale, with values greater than 0 1 in black and near 0 0 in white. A region of strong phase synchro-

nization occurs at angles near 0 0 (or equivalently near 2π) for 100 s n 150
s.6 Example data and linear analysis for time series generated from the coupled Rössler oscillators system. The coupling for 40 s t 60 s is not apparent.

(b) The WCC between sk n and rk n. The color scheme is the same as that used to create Figure 2. (c) The WC between sk n and rk n. The color scheme is the same as that used to create Figure 2.7 Computation of the embedding parameters for the time series generated from the coupled Rössler oscillators.

(a) The average mutual information (AMI) of the con- catenated sk n as a function of time lag. The first relative minimum occurs at a lag of 1 s. (b) The percentage of false nearest neighbors of the concatenated s k n as a function of the embedding dimension. The value asymptotes to a small, positive number for dimensions greater than 5.8 Synchronization measures applied to the coupled  Rössler system.

All of the measures are smoothed over  a window of size 11 at each time point. Two of the measures H x n y  and N x n y  decrease during the known interval of moderate coupling between sk n and rk n. (b) The time shifted synchronization measure T x n  η y  smoothed over a two-dimensional window of size 11 at each time point. Note that the horizontal and vertical axes show time along ensembles rk n and sk n , respectively.

The color scheme is the same as in Figure 2. The diagonal line in  the figure corresponds to the location of zero time lag. The region of maximum T x n  η y  begins approximately 4 s above the diagonal. (c) The windowed phase synchronization between s k n and rk n.

The color scheme is the same as that used to create Figure 2. An interval of weak phase synchroniza- tion occurs at angles less than 1 0 radian and near 2π radians for 40 s t 60
s.9 The ensembles of bursting data and noisy data, and the linear coupling measures. The weak oscillatory response in s k n be- tween 100 ms and 150 ms is hidden in the noise. The oscillatory response in ensemble s for 100 ms n 150 ms is apparent.

(c) The WCC between s k n and rk n. The color scheme is the same as that used to create Figure 2. (d) The WC between sk n and rk n. The color scheme is the same as that used to create Figure 2.

We find the WC is near zero for all values of frequency and time. Both linear measures detect no coupling between the two ensembles.10 Synchronization measures applied  to the burst versus  noise system. All three measures  are smoothed  over a window of size 11 ms at each time point. Both H x n y  and N x n y  fluctuate between 0 15
and 0 45 over the entire time interval
 and suggest no obvious synchronization be- tween the ensembles, as expected.

S x n y  increases during the interval 100 ms n 140 ms, and therefore suggests an increased synchronization between the en- sembles during  this interval. This incorrect interpretation is a consequence of the k increase in R x n  during the oscillatory burst, as explained in the text. (b) The  time shifted synchronization measure T x n  η y . The plotting and color scheme are the same as that used to create Figure 2.

The T x n  η y  result reveals no coupling between the ensembles. (c) The windowed phase synchronization. The plotting and color scheme are the same as that used to create Figure 2. This measure also reveals no coupling between the ensembles.11 The ensembles of bursting data and the linear coupling measures.

The weak oscillatory responses of both time series are mostly hidden in the noise. The oscillatory bursts, hidden in the single ensemble member pair of (a), are revealed here in the ensemble averaged ERPs. (c) The WCC between s k n and rk n. The color scheme is the same as that used to create Figure 2.

The WCC detects moderate cross-correlation for 130 ms t 140 ms and a time lag near
6 ms. (d) The WC between sk n and rk n. The color scheme is the same as that used to create Figure 2. The WC detect strong coherence between s k n and rk n for 100 ms t 150 ms.12 Synchronization measures applied to the ensembles of bursting data.